Inclusion Matrices and Chains

Given integers $t$, $k$, and $v$ such that $0\leq t\leq k\leq v$, let $W_{tk}(v)$ be the inclusion matrix of $t$-subsets vs. $k$-subsets of a $v$-set. We modify slightly the concept of standard tableau to study the notion of rank of a finite set of positive integers which was introduced by Frankl. Utilizing this, a decomposition of the poset $2^{[v]}$ into symmetric skipless chains is given. Based on this decomposition, we construct an inclusion matrix, denoted by $W_{\bar{t}k}(v)$, which is row-equivalent to $W_{tk}(v)$. Its Smith normal form is determined. As applications, Wilson's diagonal form of $W_{tk}(v)$ is obtained as well as a new proof of the well known theorem on the necessary and sufficient conditions for existence of integral solutions of the system $W_{tk}\bf{x}=\bf{b}$ due to Wilson. Finally we present anotherinclusion matrix with similar properties to those of $W_{\bar{t}k}(v)$ which is in some way equivalent to $W_{tk}(v)$.Comment: Accepted for publication in Journal of Combinatorial Theory, Series

On the volumes and affine types of trades

A $[t]$-trade is a pair $T=(T_+, T_-)$ of disjoint collections of subsets (blocks) of a $v$-set $V$ such that for every $0\le i\le t$, any $i$-subset of $V$ is included in the same number of blocks of $T_+$ and of $T_-$. It follows that $|T_+| = |T_-|$ and this common value is called the volume of $T$. If we restrict all the blocks to have the same size, we obtain the classical $t$-trades as a special case of $[t]$-trades. It is known that the minimum volume of a nonempty $[t]$-trade is $2^t$. Simple $[t]$-trades (i.e., those with no repeated blocks) correspond to a Boolean function of degree at most $v-t-1$. From the characterization of Kasami--Tokura of such functions with small number of ones, it is known that any simple $[t]$-trade of volume at most $2\cdot2^t$ belongs to one of two affine types, called Type\,(A) and Type\,(B) where Type\,(A) $[t]$-trades are known to exist. By considering the affine rank, we prove that $[t]$-trades of Type\,(B) do not exist. Further, we derive the spectrum of volumes of simple trades up to $2.5\cdot 2^t$, extending the known result for volumes less than $2\cdot 2^t$. We also give a characterization of "small" $[t]$-trades for $t=1,2$. Finally, an algorithm to produce $[t]$-trades for specified $t$, $v$ is given. The result of the implementation of the algorithm for $t\le4$, $v\le7$ is reported.Comment: 30 pages, final version, to appear in Electron. J. Combi

Some Indecomposable t-Designs

Abstract. The existence of large sets of 5-ð14; 6; 3Þ designs is in doubt. There are five simple 5-ð14; 6; 6Þ designs known in the literature. In this note, by the use of a computer program, we show that all of these designs are indecomposable and therefore they do not lead to large sets of 5-ð14; 6; 3Þ designs. Moreover, they provide the first counterexamples for a conjecture on disjoint t-designs which states that if there exists a t-ðv; k; lÞ design ðX; DÞ with minimum possible value of l, then there must be a t-ðv; k; lÞ design ðX; D 0 Þ such that D \ D 0 ¼ 1